Problem 42355. Minimum Set (A+A)U(A*A) OEIS A263996

Created by Richard Zapor

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{"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","targetMode":"","relationshipId":"rId1","target":"/matlab/document.xml"},{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/output","targetMode":"","relationshipId":"rId2","target":"/matlab/output.xml"}],"parts":[{"partUri":"/matlab/document.xml","relationship":[],"contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>This Challenge is to find an integer vector A that creates the minimum set size for (A+A) U (A*A) for a given vector length,</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://oeis.org/A263996\"><w:r><w:t>OEIS A263996</w:t></w:r></w:hyperlink><w:r><w:t>. The length, best value, Prime_max, and Value_max will be provided.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>The</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://oeis.org/A263996\"><w:r><w:t>OEIS A263996</w:t></w:r></w:hyperlink><w:r><w:t> gives the minimum set sizes thru length 50. Length 7 has best value 26 with Prime_max 5 and Value_max 8. A=[1 2 3 4 5 6 8] yields [1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 24 25 30 32 36 40 48 64]. The</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://68.173.157.131/Contest/SumsAndProducts1/FinalReport\"><w:r><w:t>Al Zimmermann Sums Contest Final Report</w:t></w:r></w:hyperlink><w:r><w:t> extends A263996 for lengths 40:40:1000 with complete vector solutions. The contest winner, Rokicki, noted his method used P-smooth sets, hill climbing, and random swaps. The contest was a little tougher with only L given.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Example Input/Output: L=9;Best=36;pmax=5;vmax=12; v = SP(L,Best,pmax,vmax); Yields v=[1 2 3 4 5 6 8 10 12]</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Theory/Hints: The V superset is found using</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://www.mathworks.com/matlabcentral/cody/problems/1298-p-smooth-numbers\"><w:r><w:t>psmooth(pmax,vmax)</w:t></w:r></w:hyperlink><w:r><w:t> . One observation is that for every v element the set v contains prod of all v element factor permutations. The time eater will be score evaluation. Residual evaluation suggested. A history screen, prior to score evaluation, of prior processed vectors is essential. A quick history pre-screen is vector sum. Replace testing of only values that are not factors of other numbers (eg 2,3 no replace) enables a reasonable time rolling score solution without random for the small test case values.</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

This Challenge is to find an integer vector A that creates the minimum set size for (A+A) U (A*A) for a given vector length, <a href = "https://oeis.org/A263996">OEIS A263996</a>. The length, best value, Prime_max, and Value_max will be provided.The <a href = "https://oeis.org/A263996">OEIS A263996</a> gives the minimum set sizes thru length 50. Length 7 has best value 26 with Prime_max 5 and Value_max 8. A=[1 2 3 4 5 6 8] yields [1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 24 25 30 32 36 40 48 64].
The <a href = "http://68.173.157.131/Contest/SumsAndProducts1/FinalReport">Al Zimmermann Sums Contest Final Report</a> extends A263996 for lengths 40:40:1000 with complete vector solutions. The contest winner, Rokicki, noted his method used P-smooth sets, hill climbing, and random swaps. The contest was a little tougher with only L given.Example Input/Output:
L=9;Best=36;pmax=5;vmax=12;
v = SP(L,Best,pmax,vmax); Yields v=[1 2 3 4 5 6 8 10 12]Theory/Hints: The V superset is found using <a href = "http://www.mathworks.com/matlabcentral/cody/problems/1298-p-smooth-numbers">psmooth(pmax,vmax)</a> . One observation is that for every v element the set v contains prod of all v element factor permutations. The time eater will be score evaluation. Residual evaluation suggested. A history screen, prior to score evaluation, of prior processed vectors is essential. A quick history pre-screen is vector sum. Replace testing of only values that are not factors of other numbers (eg 2,3 no replace) enables a reasonable time rolling score solution without random for the small test case values.

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Problem 42355. Minimum Set (A+A)U(A*A) OEIS A263996

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